Determinantal Representations of the Core Inverse and Its Generalizations with Applications

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.


Introduction
In the whole article, the notations R and C are reserved for fields of the real and complex numbers, respectively.C m×n stands for the set of all m × n matrices over C. C m×n r determines its subset of matrices with rank r.For A ∈ C m×n , the symbols A T , A * , and rk(A) specify the transpose, the conjugate transpose, and the rank of A, respectively.|A| or detA stands for its determinant.A matrix A ∈ C n×n is Hermitian if A * � A.
A † means the Moore-Penrose inverse of A ∈ C n×m , i.e., the exclusive matrix X satisfying the following four equations: (AX) * � AX, (XA) * � XA.
For A ∈ C n×n with index IndA � k, i.e., the smallest positive number such that rk(A k+1 ) � rk(A k ), the Drazin inverse of A, denoted by A d , is called the unique matrix X that satisfies equation (2) and the following equations: XA k+1 � A k , (6a) In particular, if IndA � 1, then the matrix X is called the group inverse and it is denoted by X � A # .If IndA � 0, then A is nonsingular and It is evident that if the condition ( 5) is fulfilled, then (6a) and (6b) are equivalent.We put both these conditions because they will be used below independently of each other and without the obligatory fulfillment of (5).
A matrix A satisfying the conditions (i), (j), . . . is called an i, j, . . . -inverse of A and is denoted by A (i,j,...) .e set of matrices A (i,j,...) is denoted by A i, j, . . . .In particular, A (1) is called the inner inverse of A, A (2) is called the outer inverse of A, A (1,2) is called the reflexive inverse of A, A (1,2,3,4) is its Moore-Penrose inverse, etc.
For an arbitrary matrix A ∈ C m×n , we denote by (i) N(A) � x ∈ C n×1 : Ax � 0  , the kernel (or the null space) of A (ii) C(A) � y ∈ C m×1 : y � Ax, x ∈ C n×1  , the column space (or the range space) of A (iii) R(A) � y ∈ C 1×n : y � xA, x ∈ C 1×m  , the row space of A P A ≔ AA † and Q A ≔ A † A are the orthogonal projectors onto the range of A and the range of A * , respectively.
e core inverse was introduced by Baksalary and Trenkler in [1].Later, it was investigated by Xu et al. in [2], among others.Rakić et al. in [3] generalized the core inverse of a complex matrix to the case of an element in a ring.
In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, for generalized inverse matrices there exist different determinantal representations as a result of the search of their more applicable explicit expressions (see, e.g., [19][20][21][22][23][24][25]).In this paper, we get new determinantal representations of the core inverse and its generalizations by using the author's recently obtained determinantal representations of the Moore-Penrose inverse and the Drazin inverse over the quaternion skew field and the field of complex numbers as a special case [26,27].
Note that a determinantal representation of the core-EP generalized inverse in complex matrices has been derived in [4] based on the determinantal representation of a reflexive inverse obtained in [19,20].
One of possible applications of the determinantal representations of the inverse matrix is Cramer's rule to find solutions of systems of linear equations.Determinantal representations of generalized inverses have similar applications.In this paper, we derive the determinantal representation of the Bott-Duffin inverse that has close relation with the core inverse, and apply it to obtain Cramer's rule for the constrained linear equations.
e paper is organized as follows.In Section 2, we start with preliminary introduction of determinantal representations of the Moore-Penrose inverse and the Drazin inverse.In the next sections, we give determinantal representations of the core inverse and its generalizations.In particular, determinantal representations of the right and left core inverses are established in Section 3, of the right and left core-EP inverses in Section 4, and of the DMP inverse and its dual MPD inverse in Section 5. Determinantal representations of the CMP inverse are obtained in Section 6.In Section 7, we derive Cramer's rule for the constrained linear equations by using the determinantal representation of the Bott-Duffin inverse that is same as for the right core inverse.A numerical example to illustrate the main results is considered in Section 8. Finally, in Section 9, the conclusions are drawn.

Preliminaries
e submatrix of A ∈ C m×n with rows and columns indexed by α and β, respectively, and denoted by A α β .en, A α α is a principal submatrix of A with rows and columns indexed by α, and |A| α α is the corresponding principal minor of the determinant |A|.Suppose that stands for the collection of strictly increasing sequences of 1 ≤ k ≤ n integers chosen from 1, . . ., n { }.For fixed i ∈ α and Denote by a .jand a * .j, a i. and a * i. the jth columns and the ith rows of A and A * , respectively.By A i. (b) and A .j (c), we denote the matrices obtained from A by replacing its ith row with the row b and its jth column with the column c.
Theorem 1 [21].en, the following determinantal representations can be obtained: where _ a .j is the jth column and _ a i. is the ith row of where a (k)  i. is the ith row and a (k)  .j is the jth column of A k .

Determinantal Representations of the Core Inverses
Together with the core inverse in [3], it was introduced the dual core inverse.Since the both these core inverses are equipollent and they are different only in the position relative to the inducting matrix A, we propose to call them as the right and left core inverses regarding to their positions.So, according to [1], we have the following definition that is equivalent to Definition 1.
Definition 2. A matrix X ∈ C n×n is said to be the right core inverse matrix of A ∈ C n×n if it satisfies the conditions When such matrix X exists, it is denoted by A ○ # .e following definition of the left core inverse can be given that is equivalent to the introduced dual core inverse [3].Definition 3. A matrix X ∈ C n×n is said to be the left core inverse matrix of A ∈ C n×n if it satisfies the conditions When such matrix X exists, it is denoted by A ○ # .
Remark 2. In [42], the conditions of the dual core inverse are given as follows: , then these conditions and ( 17) are analogous.Remark 3. In Definitions 2 and 3, we purposely state that these definitions concern with matrices, since the right and core inverses in a ring were introduced recently in [43].e notions of the right and left core inverse matrices, introduced here, and of the right and left core inverses in a ring, introduced in [43], have no direct connection with each other.
According to [1], we introduce the following sets of matrices: e matrices from C CM n are called group matrices or core matrices.
Moreover, if A is nonsingular, IndA � 0, then its core inverses are the usual inverse.According to [1], we have the following representations of the right and left core inverses.

Lemma 2 [1]. Let
Remark 4. In eorems 2 and 3, we will suppose that and A ∈ C EP n (in particular, A is Hermitian), then from Lemma 2 and the definitions of the Moore-Penrose and group inverses it follows that en, its right core inverse matrix A ○ # � (a ○ #,r ij ) has the following determinantal representations: (1)   i. (2)   .j

􏼐 􏼑 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌
where u (2)  .j� are the row and column vectors, respectively.Here,  a .fand  a l.

are the fth column and lth row of 􏽥
Proof.Taking into account (20), we have for By substituting ( 15) and ( 12) in ( 25), we obtain where e .land e l. are the unit column and row vectors, respectively, such that all their components are 0, except the lth components which are 1;  a lf is the (lf)th element of the matrix  A ≔ A 2 A * .Let u (1)  il ≔ Construct the matrix AA *  j. u (1)   i.

□
Taking into account (21), the following theorem on the determinantal representation of the left core inverse can be proved similarly. (1)  i. (2)   .j

􏼐 􏼑 􏼌 􏼌 􏼌 􏼌 􏼌 Journal of Mathematics where v (1)  i. � Here, a .fand a l. are the fth column and lth row of

Determinantal Representations of the Core-EP Inverses
Similar to [4], we introduce two core-EP inverses.
Definition 4. A matrix X ∈ C n×n is said to be the right core-EP inverse of A ∈ C n×n if it satisfies the conditions It is denoted by A ○ † .Definition 5. A matrix X ∈ C n×n is said to be the left core-EP inverse of A ∈ C n×n if it satisfies the conditions It is denoted by [4], and the dual core-EP inverse introduced in [42].
According to [4], we have the following representations the core-EP inverses of A ∈ C n×n : anks to [42], the following representations of the core-EP inverses will be used for their determinantal representations.

Lemma 3. Let
Moreover, if IndA � 1, then we have the following representations of the right and left core inverse matrices: Theorem 4. Suppose A ∈ C n×n , IndA � k, and rkA k � s, and there exist where Taking into account (10) for the determinantal representation of (A k+1 ) † , we get where a (k+1) t.

□
Taking into account the representations (38)- (39), we derive the determinantal representations of the right and left core inverse matrices that have simpler expressions than those obtained in eorems 2 and 3.

Corollary 3.
Let A ∈ C n×n s and IndA � 1, and there exist A ○ # and A ○ # .en, A ○ # � (a ○ #,r ij ) and A ○ # � (a ○ #,l ij ) can be expressed as follows: where  a i . is the ith row of  A � A(A 2 ) * and � a .j is the jth column of � A � (A 2 ) * A.

Determinantal Representations of the DMP and MPD Inverses
e concept of the DMP inverse in complex matrices was introduced in [6] by Malik and ome.Definition 6. [6] Suppose A ∈ C n×n and IndA � k.A matrix X ∈ C n×n is said to be the DMP inverse of A if it satisfies the conditions It is denoted by A d, † .According to [6], if an arbitrary matrix satisfies the system of equations (45), then it is unique and has the following representation: Theorem 5. Let A ∈ C n×n s , IndA � k, and rk(A k ) � s 1 .en, its DMP inverse A d, † � (a d, † ij ) has the following determinantal representations: (1)   i.

□
e name of the DMP inverse is in accordance with the order of using the Drazin inverse (D) and the Moore-6 Journal of Mathematics Penrose (MP) inverse.In that connection, it would be logical to consider the following definition.
Definition 7. Suppose A ∈ C n×n and IndA � k.A matrix X ∈ C n×n is said to be the MPD inverse of A if it satisfies the conditions It is denoted by A †,d .Similar as for the DMP inverse, it can be proved that the matrix A †,d is unique, and it can be represented as Theorem 6.Let A ∈ C n×n s , IndA � k, and rkA k � s 1 .en, its MPD inverse A †,d � (a †,d ij ) has the following determinantal representations: (1)   .j (2)  i.

􏼐 􏼑 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌 􏼌
where v (1)  .j� Here,  a l. and  a .fare the lth row and the fth column of Proof.
e proof is similar to the proof of eorem 6. □

Determinantal Representations of the CMP Inverse
Definition 8 [7].Suppose A ∈ C n×n has the core-nilpotent decomposition A � A 1 + A 2 , where IndA 1 � IndA, A 2 is nilpotent, and Lemma 4 [7].Let A ∈ C n×n .e matrix X � A c, † is the unique matrix that satisfies the following system of equations: Taking into account (60), it follows the next theorem about determinantal representations of the quaternion CMP inverse.

Theorem 7.
Let A ∈ C n×n s , IndA � m, and rk(A m ) � s 1 .en, the determinantal representations of its CMP inverse A c, † � (a c, † ij ) can be expressed as .j

􏼐 􏼑 􏼌 􏼌 􏼌 􏼌 􏼌 for all l � 1, 2, where v (1)  .j� w (1)  i. � v (2)  .j� (a) Taking into account the expressions ( 14), (12), and ( 13) for the determinantal representations of A d , Q A , and P A , respectively, we have where _ a .t is the tth column of A * A, € a l. is the lth row of AA * , and a (m)   t.
is the tth row of A m .So, it is clear that where e .t is the tth unit column vector, e k. is the kth unit row vector, and  a tk is the (tk)th element of the tth component of a column-vector then Construct the matrix U � (u tl ) ∈ C n×n , where u tl is given by (70), and denote  U ≔ UAA * .en, taking into account that If we put that is the tth component of a column vector, v (1)  .j� [v (1)  1j , . . ., v (1)  nj ]. en from (1)   .j it follows (61) with v (1)  .jgiven by (62).If we initially put Journal of Mathematics as the kth component of the row vector, w (1)  i. � [w (1)  i1 , . . ., w (1)  in ]. en from A 2   j. w (1)   i.
□ Remark 6. Concerning the possible existence of dual to the CMP inverse, we note the following.Taking into account (88)

Cramer's Rule for Some Constrained Linear Equations
Cramer's rules for special solutions to systems of linear equations stand as possible applications of determinantal representations of the core inverse and its generalizations.For instance, consider Cramer's rule for the system which has the important applied significance.First, note that the core inverse has close relation with the Bott-Duffin inverse.According to [19], the Bott-Duffin inverse of A ∈ H n×n with respect to C(A) can be given by where (A − I n )P A + I n needs to be nonsingular.According to [1], A (− 1) C(A) coincides with the right core inverse A ○ # .One can find more details regarding the Bott-Duffin inverse in [44].
Consider the constrained linear equations where A ∈ C n×n and b ∈ C n .According to [45], this equation arises in electrical networks and its solution is determined by the following lemma.
Lemma 5. Let (A − I n )P A + I n be nonsingular.en, equation ( 90) has for every b the unique solution e following theorem gives Cramer's rule of finding the solution (91).

□
It is evident that the solution y � [y 1 , . . ., y n ] T from (92) by the components can be expressed as where  x i is the ith component of the column vector  x � Ax.

Conclusions
In this chapter, we get the direct method to find of the core inverse and its generalizations that is based on their determinantal representations.New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are Journal of Mathematics derived.Determinantal representation the Bott-Duffin inverse and its application to get Cramer's rule of the solution to the constrained linear equations are obtained.

is the ith row and € a .j is the jth column of AA * . e following lemma gives determinantal representa- tions of the Drazin inverse in complex matrices. Lemma 1 [22]. Let A ∈
For the projector PA � (p ij ) m×m , C n×n with IndA � k and rkA k+1 � rkA k � r.
Here,  u t. is the tth row and  u .k is the kth column of  U ≔ UAA * ,  g t. is the tth row and  g .k is the kth column of Journal of Mathematics  G ≔ A * AG, and the matrices U � (u ij ) ∈ C n×n and G � (g ij ) ∈ C n×n are such that u ij � Suppose b � [b 1 , . . ., b n ] T and x � [x 1 , . . ., x n ] T .